Quantum Speedups of Optimizing Approximately Convex Functions with Applications to Logarithmic Regret Stochastic Convex Bandits

TL;DR

This paper introduces quantum algorithms that significantly speed up the optimization of approximately convex functions and applies these methods to improve regret bounds in stochastic convex bandit problems.

Contribution

The paper develops the first quantum algorithms for approximately convex function optimization and demonstrates exponential and polynomial speedups in relevant parameters.

Findings

Quantum algorithms find near-optimal solutions with fewer evaluations.

Achieves polynomial quantum speedup over classical methods in function evaluation.

Provides exponential speedup in regret bounds for stochastic convex bandits.

Abstract

We initiate the study of quantum algorithms for optimizing approximately convex functions. Given a convex set ${\cal K}\subseteq\mathbb{R}^{n}$ and a function $F\colon\mathbb{R}^{n}\to\mathbb{R}$ such that there exists a convex function $f\colon\mathcal{K}\to\mathbb{R}$ satisfying $\sup_{x\in{\cal K}}|F(x)-f(x)|\leq \epsilon/n$, our quantum algorithm finds an $x^{*}\in{\cal K}$ such that $F(x^{*})-\min_{x\in{\cal K}} F(x)\leq\epsilon$ using $\tilde{O}(n^{3})$ quantum evaluation queries to $F$. This achieves a polynomial quantum speedup compared to the best-known classical algorithms. As an application, we give a quantum algorithm for zeroth-order stochastic convex bandits with $\tilde{O}(n^{5}\log^{2} T)$ regret, an exponential speedup in $T$ compared to the classical $\Omega(\sqrt{T})$ lower bound. Technically, we achieve quantum speedup in $n$ by exploiting a quantum framework of simulated annealing and adopting a quantum version of the hit-and-run walk. Our speedup in $T$ for zeroth-order stochastic convex bandits is due to a quadratic quantum speedup in multiplicative error of mean estimation.